"Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning."

The above quote comes from the National Council of Teachers of Mathematics' (NCTM) discussion of the Technology Principle in the Principles and Standards for School Mathematics (PSSM). NCTM also states:

Electronic technologies - calculators and computers - are essential tools for teaching, learning, and doing mathematics. They furnish visual images of mathematical ideas ... and they compute efficiently and accurately ... Students can learn more mathematics more deeply with the appropriate use of technology (Dunham and Dick 1994; Sheets 1993; Boers-van Oosterum 1990; Rojano 1996; Groves 1994). Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions ... In short, technology can help students learn mathematics.
(Professional Standards for Teaching Mathematics, 1991, p. 24-25)

Both Janel Green and Tom Reardon teach lessons in which technology is essential in helping to foster students' understanding and use of intuition, and to focus on decision making, reflection, and reasoning. When used appropriately, technology can enhance students' learning opportunities and connect the development of skills and procedures to the more general development of mathematical understanding.

Listen to what Diane Briars has to say about Tom Reardon's use of technology in the video for Workshop 2, Part I:

One of the most impressive things about Tom's class was his use of technology...
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Fostering Understanding and Use of Intuition

Teaching is not telling, and an excellent way to help students understand mathematical concepts is to use technology appropriately so that they can look at relationships, draw conclusions, and reach an understanding on their own that is deep and meaningful. "With calculators and computers students can examine more examples or representational forms than are feasible by hand, so they can make and explore conjectures easily," according to NCTM. "The graphic power of technological tools affords access to visual models that are powerful but that many students are unable to generate independently." (PSSM, 2000)

In Tom Reardon's class, the students were able to use technology to see the connections between the parameters in their function and the resulting graph and table. The slope and y-intercept weren't just numbers in an equation but had a meaning, and this meaning became apparent in the table of values and the graph.

Read what Janel Green says about the importance of understanding different methods for solving equations:

The key concept was for students to relate making a table and solving algebraically and making a graph to solve any problem...
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Reflection:

Think about a lesson you teach that uses technology to foster understanding and intuition in your students. Describe several ways that the activities in this lesson demonstrate an appropriate use of technology.

Focusing on Decision Making, Reflection, and Reasoning

"Using technological tools, students can reason about more general issues, such as parameter changes, and they can model and solve complex problems that were heretofore inaccessible to them," according to NCTM (PSSM, 2000). Tom Reardon encouraged decision making in his class when he led a discussion of the different ways students chose to solve the problems. Some of the students were substituting values into the equation, some were looking for values in the table, and some used a graph. He then asked them to verify that all the methods produced the same solution. Janel Green encouraged her students not only to learn all three methods for solving equations and inequalities but also to reflect on which method might be most efficient in a given situation.

In Janel's class, technology was helpful because students were learning how to solve equations and inequalities using tables, graphs and algebra for the first time. Later on in their mathematical studies, however, it might be considered inappropriate to use a calculator to solve these problems; students should be expected to find solutions mentally. Therefore, teachers must make sure students improve their mental math skills when using technology. Students need to be able to examine an electronic graph or table, for example, and determine that it makes sense and is reasonable, and they should be able to explain thoughtfully why a solution generated by a calculator is correct or incorrect.

Read what Fran Curcio has to say about how calculator use can help students reason about the important aspects of a problem:

I would try to highlight what we did in numbers and in the words that went with it, so that they would see a subtle, subliminal suggestion...
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Reflection:

Reflect on a lesson you teach and describe how you can use technological tools to help students reason about the lesson's general issues and concepts.

Enhancing Students' Learning Opportunities

According to NCTM, "Teachers should use technology to enhance their students' learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well - graphing, visualizing, and computing." (PSSM, 2000) Janel Green used a familiar problem context and graphing calculators to support her students in reaching an important and difficult mathematical goal: the ability to solve equations and inequalities using tables, graphs, and algebra. Tom Reardon used a variety of technologies and a seemingly more complex problem to help his students deepen their understanding of linear functions and how slopes and y-intercepts relate to real-world situations.

Read what Tom Reardon says about how the SMART Board has helped enhance his students' learning opportunities, even when they've been absent from class:

Whatever I do on that SMART Board, I can save it as an HTML file and I can put it on my Web site with the date.
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Reflection:

What types of lessons do you use that incorporate technology as a means to enhance student learning opportunities? How would those learning opportunities differ if technology wasn't used?

Connecting Skills and Procedures to Mathematical Understanding

"Technology can help teachers connect the development of skills and procedures to the more general development of mathematical understanding," according to NCTM. "As some skills that were once considered essential are rendered less necessary by technological tools, students can be asked to work at higher levels of generalization or abstraction." (PSSM, 2000) Because Tom Reardon used a contextual problem and incorporated technology, he was able to help his students understand the general idea that the cost of the phone bill was a function of the length of the phone call. He was able to move naturally between different representations of the linear function that the class used to model the problem. They were able to understand that y = 0.24x + 0.85, C = 0.24t + 0.85, and C(t) = 0.24t + 0.85 were all equivalent representations for the problem.

Read what Janel Green says about how her students began to see the connections between the numeric, graphical, and algebraic solutions:

Sometimes technology allows us to do things in mathematics that we couldn't do without it...
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In the Hot Dog Sales lesson featured in Workshop 2, Janel Green works with her students to find the break-even point, where revenue and costs are equal. She shows them how to solve the equation they generated, 0.50x - 450 = 0, using tables, graphs, and algebra. Then she has her students work in groups and apply and extend these concepts. Even though this was the first time these students had ever been exposed to these ideas, many of them were able to figure out how to use what they had learned about equations and to transfer that information to finding the solution of inequalities. Having the ability to solve inequalities numerically and graphically helped the students in their struggle to figure out the algebraic solution.

Below is an example of how one group of students found the solution to one of the inequality questions in the Hot Dog Sales lesson. The students were asked to show the number of hot dogs that need to be sold to produce a profit of at least $250.

This group of students needed Janel's help to figure out the correct algebraic solution strategy.

A very important concept for students to understand is that both the numerical method (using tables) and the graphical method are general methods of solutions and can be used to solve equations and inequalities of any function. In contrast, the algebraic method of solution only works for linear equations and inequalities. Students must learn the particular algebraic manipulation needed to solve each type of function. In fact, solving inequalities algebraically is extremely difficult, and sometimes impossible, for any function other than a linear function.

Reflection:

Write about a lesson you teach that helps students develop particular skills and procedures as well as a more general mathematical understanding.

Cost Considerations

Some schools and teachers believe that the cost of technology prohibits them from using it effectively. However, there are several ways to make sure that your classroom has the appropriate technology.

Listen to what Tom Reardon says about finding ways to purchase appropriate technology: